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Rank Minimization with Structured Data Patterns

  • Viktor Larsson
  • Carl Olsson
  • Erik Bylow
  • Fredrik Kahl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8691)

Abstract

The problem of finding a low rank approximation of a given measurement matrix is of key interest in computer vision. If all the elements of the measurement matrix are available, the problem can be solved using factorization. However, in the case of missing data no satisfactory solution exists. Recent approaches replace the rank term with the weaker (but convex) nuclear norm. In this paper we show that this heuristic works poorly on problems where the locations of the missing entries are highly correlated and structured which is a common situation in many applications.

Our main contribution is the derivation of a much stronger convex relaxation that takes into account not only the rank function but also the data. We propose an algorithm which uses this relaxation to solve the rank approximation problem on matrices where the given measurements can be organized into overlapping blocks without missing data. The algorithm is computationally efficient and we have applied it to several classical problems including structure from motion and linear shape basis estimation. We demonstrate on both real and synthetic data that it outperforms state-of-the-art alternatives.

Keywords

Singular Value Decomposition Rank Function Measurement Matrix Convex Relaxation Convex Envelope 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Viktor Larsson
    • 1
  • Carl Olsson
    • 1
  • Erik Bylow
    • 1
  • Fredrik Kahl
    • 1
    • 2
  1. 1.Centre for Mathematical SciencesLund UniversitySweden
  2. 2.Department of Signals and SystemsChalmers University of TechnologySweden

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